The mathematically rigorous analysis of this equation goes back to
Wald [1]} and Kay [2]}, who established uniform boundedness of solutions.
In the spirit of the positive commutator methods outlined above, Dafermos and Rodniansk [3]} found a robust approach based on carefully chosen vector fields and multipliers. See Luk's work [4]}, [5]} which is in a similar spirit. As already noted, Blue and Soffer [6]} proved local decay estimates using Morawetz estimates. Dafermos and Rodnianski [7]} proved Price's \(t^{-3}\)  decay law for a nonlinear problem but assuming spherical symmetry.
